Page:A History of Mathematics (1893).djvu/283

 former method was developed by Ferrari, Vieta, Tchirnhausen, Euler, Bézout, and Lagrange; the latter by Vandermonde and Lagrange.[20] In the method of substitution the original forms are so transformed that the determination of the roots is made to depend upon simpler functions (resolvents). In the method of combination auxiliary quantities are substituted for certain simple combinations ("types") of the unknown roots of the equation, and auxiliary equations (resolvents) are obtained for these quantities with aid of the coefficients of the given equation. Lagrange traced all known algebraic solutions of equations to the uniform principle consisting in the formation and solution of equations of lower degree whose roots are linear functions of the required roots, and of the roots of unity. He showed that the quintic cannot be reduced in this way, its resolvent being of the sixth degree. His researches on the theory of equations were continued after he left Berlin. In the Résolution des équations numériques (1798) he gave a method of approximating to the real roots of numerical equations by continued fractions. Among other things, it contains also a proof that every equation must have a root,—a theorem which appears before this to have been considered self-evident. Other proofs of this were given by Argand, Gauss, and Cauchy. In a note to the above work Lagrange uses Fermat's theorem and certain suggestions of Gauss in effecting a complete algebraic solution of any binomial equation.

While in Berlin Lagrange published several papers on the theory of numbers. In 1769 he gave a solution in integers of indeterminate equations of the second degree, which resembles the Hindoo cyclic method; he was the first to prove, in 1771, "Wilson's theorem," enunciated by an Englishman, John Wilson, and first published by Waring in his Meditationes Algebraicœ; he investigated in 1775 under what conditions $$\scriptstyle{\pm 2}$$ and $$\scriptstyle{\pm 5}$$ ($\scriptstyle{-1}$ and $$\scriptstyle{\pm 3}$$ having been discussed by Euler)